Three-region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate

TL;DR

This paper derives a three-region inequality for second order elliptic equations with discontinuous coefficients, using Carleman estimates, and applies it to size estimation problems with limited boundary data.

Contribution

The paper introduces a novel three-region inequality for elliptic equations with jump discontinuities, extending quantitative uniqueness results.

Findings

Established a three-region inequality for elliptic equations with discontinuous coefficients

Applied the inequality to size estimation with minimal boundary measurements

Enhanced understanding of unique continuation properties in complex media

Abstract

In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequality, for the second order elliptic equation with jump discontinuous coefficients. The derivation of the inequality relies on the Carleman estimate proved in our previous work. We then apply the three-region inequality to study the size estimate problem with one boundary measurement.